On the Lorentz surfaces

This thesis consists of three chapters. In the first chapter; the basic notions which are defined by a symmetric bilinear form on a vector space are given. The basic notions for Riemannian manifold which is defined by positive definite metrics and Lorentzian manifold which is defined by indefinite metrics are introduced. In the second chapter; global properties of Lorentzian metrics are studied. In the third chapter; firstly Euclidean planes and Minkowski planes are considered. Then, local surfaces on the Euclidean-3 spaces and Minkowski-3 spaces are mutually examined. The end of the examination, results indicated the relations between Euclidean and Minkowski invaryant (Gauss and mean curvatures, fundamental forms,...) which compose this local surface are given. KEY WORDS: Lorentzian manifold, Riemannian manifold, indefinite metric, Euclidean plane, Minkowski plane, local surface, Euclidean-3 space, Minkowski-3 space 11